So I'm having some problems understanding how if a moving object emits light, the speed of that light is still the constant, $c$, speed of light. So according to Einstein's theory of relativity, light travels the at the same speed for all inertial frames of reference. So from what I understand if a driver driving a car at a constant velocity with headlights on sees the light, it would be travelling at $c$, and an observer on the side of the road would also see the light as speed $c$. At school we have learn about time dilation and length contraction however I do not see how they could be applied in this situation to explain why the light "slows down". I would appreciate it if your answers could try to remain with a knowledge level of a high schooler.
I have also looked at other threads on this exchange, but either I do not understand them or they don't answer the question I have.